Momentum: The Navier-Stokes Equations

2.5 Conservation of Momentum: The Navier-Stokes Equations of Motion

2.5.1 Cartesian Coordinates

Application of Newton’s law of motion to the

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element shown in Fig. 2.5, gives

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Application of (a) in the x-direction, gives

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Each term in (b) is expressed in terms of flow

field variables: density, pressure, and velocity components:

Mass of the element:

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IMPORTANT

THE NORMAL AND SHEARING STRESSES ARE EXPRESSED IN TERMS

OF VELOSICTY AND PRESSURE. THIS IS VALID FOR NEWTOINAN

FLUIDS. (See equations 2.7a-2.7f).

THE RESULTING EQUATIONS ARE KNOWN AS THE NAVIER-STOKES

EQUAITONS OF MOTION

SPECIAL SIMPLIFIED CASES:

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2.5.2 Cylindrical Coordinates

The three equations corresponding to (2.10) in cylindrical coordinates are (2.11r), clip_image016

and (2.11z).

2.5.3 Spherical Coordinates

The three equations corresponding to (2.10) in spherical coordinates are (2.11r), clip_image016[1]and clip_image016[2]

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